 Negative curves of a big real divisor lie in finitely many divisors

Write a big real Cartier divisor on an integral projective variety as $D{\sim }_{\mathbb{R}}B+E$, with $B$ ample and $E$ effective real Cartier. Every curve outside $\mathrm{Supp}E$ has $D\cdot C=B\cdot C+E\cdot C>0$. Therefore curves negative against $D$ lie in finitely many support components of codimension one. On a projective scheme use componentwise bigness on a projective scheme and collect the supports on its finitely many integral components; codimension one is taken in the relevant component.